The Groups G2n with Additional Structures

Abstract

In the paper [1], V. O. Manturov introduced the groups G^k_n depending on two natural parameters n > k and naturally related to topology and to the theory of dynamical systems. The group G²_n, which is the simplest part of G^k_n, is isomorphic to the group of pure free braids on n strands. In the present paper, we study the groups G²_n supplied with additional structures–parity and points; these groups are denoted by G²_n,p and G²_n,d. First,we define the groups G²_n,p and G²_n,d, then study the relationship between the groups G²_n, G²_n,p, and G²_n,d. Finally, we give an example of a braid on n + 1 strands, which is not the trivial braid on n + 1 strands, by using a braid on n strands with parity. After that, the author discusses links in S_g × S¹ that can determine diagrams with points; these points correspond to the factor S1 in the product S_g × S¹.